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2013, 3(4): 655-664. doi: 10.3934/naco.2013.3.655

Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method

Honglei Xu 1, , Peng Sui 2, , Guanglu Zhou 3, and  Louis Caccetta 3,

1. 

School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China
2. 

Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China
3. 

Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia, Australia

Received  March 2013 Revised  October 2013 Published  October 2013

In this paper, we consider the bullwhip effect problem of an Order-Up-To (OUT) inventory strategy for a supply chain system. We firstly establish a new discrete-time dynamical model which is suitable to describe the OUT inventory strategy. Then, we analyze the bullwhip effect for the dynamical model of the supply chain system. We thus transform the bullwhip effect's dampening problem to a discrete-time optimal control problem. By using the Pontryagin's maximum principle, we compute the corresponding optimal control and obtain the optimal manufacturer productivity of goods. Finally, we carry out numerical simulation experiments to show that the devised optimal control strategy is useful to dampen the bullwhip effect which always happens in the supply chain system.
Keywords: optimal control., order-up-to inventory strategy, Bullwhip effect.
Mathematics Subject Classification: Primary: 90B50; Secondary: 65P9.
Citation: Honglei Xu, Peng Sui, Guanglu Zhou, Louis Caccetta. Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 655-664. doi: 10.3934/naco.2013.3.655
References:
[1]

R. B. Chase and N. J. Aquilano, "Production and Operations Management,", Irwin. 7th ed, (1995). Google Scholar

[2]

F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information,, Management Science, 46 (2000), 436. Google Scholar

[3]

Y. F. Chen and S. M. Disney, The myopic order-up-to policy with a proportional feedback controller,, Int J Prod Res, 45 (2007), 351. Google Scholar

[4]

S. Eiamkanchanalai and A. Banerjee, Production lot sizing with variable production rate and explicit idle capacity cost,, International Journal of Production Economics, 59 (1999), 251. Google Scholar

[5]

J. Forrestor, "Industrial Dynamics,", New York: MIT Press and John Wily &Sons, (1961). Google Scholar

[6]

L. L. Hau, V. Padmanabhan and S. Whang, Information distortion in a supply chain: the bullwhip effect,, Management Science, 43 (1997), 546. Google Scholar

[7]

S. Karlin and H. Scarf, "One Stage Inventory Models with Uncertainty. In: Studies in the Mathematical Theory of Inventory and Production,", Stanford University Press, (1958), 109. Google Scholar

[8]

S. Makridakis, S. C. Wheelwright and V. E. McGee, "Forecasting: Methods and Applications,", John Wiley & Sons, (1978). Google Scholar

[9]

E. Ricard and K. Baradia, Evaluation of supply chain structures through modularization and postponement,, European Journal Of Operational Research, 124 (2000), 495. Google Scholar

[10]

J. K. Ryan, "Analysis of Inventory Models with Limited Demand Information,", Ph.D. Dissertation, (1997). Google Scholar

[11]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory-Applications to Management Science,", Martinus Nijhoff, (1981). Google Scholar

[12]

J. D. Sterman, Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment,, Management Science, 35 (1989), 321. Google Scholar

[13]

K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems,", Longman Scientific & Technical, (1991). Google Scholar

[14]

D. R. Towill and M. M. Naim, Industrial dynamics simulation models in the design of supply chains,, International Journal of Physical Distribution and Logistics Management, 22 (1992), 3. Google Scholar

[15]

K. F. C. Yiu, L. L. Xie and K. L. Mak, Analysis of bullwhip effect in supply chains with heterogeneous decision models,, Journal of Industrial and Management Optimization, 2 (2009), 81. doi: 10.3934/jimo.2009.5.81. Google Scholar

show all references

References:
[1]

R. B. Chase and N. J. Aquilano, "Production and Operations Management,", Irwin. 7th ed, (1995). Google Scholar

[2]

F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information,, Management Science, 46 (2000), 436. Google Scholar

[3]

Y. F. Chen and S. M. Disney, The myopic order-up-to policy with a proportional feedback controller,, Int J Prod Res, 45 (2007), 351. Google Scholar

[4]

S. Eiamkanchanalai and A. Banerjee, Production lot sizing with variable production rate and explicit idle capacity cost,, International Journal of Production Economics, 59 (1999), 251. Google Scholar

[5]

J. Forrestor, "Industrial Dynamics,", New York: MIT Press and John Wily &Sons, (1961). Google Scholar

[6]

L. L. Hau, V. Padmanabhan and S. Whang, Information distortion in a supply chain: the bullwhip effect,, Management Science, 43 (1997), 546. Google Scholar

[7]

S. Karlin and H. Scarf, "One Stage Inventory Models with Uncertainty. In: Studies in the Mathematical Theory of Inventory and Production,", Stanford University Press, (1958), 109. Google Scholar

[8]

S. Makridakis, S. C. Wheelwright and V. E. McGee, "Forecasting: Methods and Applications,", John Wiley & Sons, (1978). Google Scholar

[9]

E. Ricard and K. Baradia, Evaluation of supply chain structures through modularization and postponement,, European Journal Of Operational Research, 124 (2000), 495. Google Scholar

[10]

J. K. Ryan, "Analysis of Inventory Models with Limited Demand Information,", Ph.D. Dissertation, (1997). Google Scholar

[11]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory-Applications to Management Science,", Martinus Nijhoff, (1981). Google Scholar

[12]

J. D. Sterman, Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment,, Management Science, 35 (1989), 321. Google Scholar

[13]

K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems,", Longman Scientific & Technical, (1991). Google Scholar

[14]

D. R. Towill and M. M. Naim, Industrial dynamics simulation models in the design of supply chains,, International Journal of Physical Distribution and Logistics Management, 22 (1992), 3. Google Scholar

[15]

K. F. C. Yiu, L. L. Xie and K. L. Mak, Analysis of bullwhip effect in supply chains with heterogeneous decision models,, Journal of Industrial and Management Optimization, 2 (2009), 81. doi: 10.3934/jimo.2009.5.81. Google Scholar

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